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Meanfield analysis of Markov models with reward feedback
"... Abstract. We extend the population continuous time Markov chain formalism so that the state space is augmented with continuous variables accumulated over time as functions of component populations. System feedback can be expressed using accumulations that in turn can influence the Markov chain behav ..."
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Abstract. We extend the population continuous time Markov chain formalism so that the state space is augmented with continuous variables accumulated over time as functions of component populations. System feedback can be expressed using accumulations that in turn can influence the Markov chain behaviour via functional transition rates. We show how to obtain meanfield differential equations capturing means and higherorder moments of the discrete populations and continuous accumulation variables. We also provide first and secondorder convergence results and suggest a novel normal moment closure that can greatly improve the accuracy of means and higher moments. We demonstrate how such a framework is suitable for modelling feedback from globallyaccumulated quantities such as energy consumption, cost or temperature. Finally, we present a worked example modelling a hypothetical heterogeneous computing cluster and its interaction with air conditioning units. 1
Fluid computation of the performance–energy tradeoff in large scale Markov models
"... Recent fluid analysis techniques allow fast and efficient calculation of complex reward metrics and passage time probabilities in systems with very large state space. We demonstrate how to incorporate these to look at the tradeoff between service level agreement (SLA) satisfaction and complex rewar ..."
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Recent fluid analysis techniques allow fast and efficient calculation of complex reward metrics and passage time probabilities in systems with very large state space. We demonstrate how to incorporate these to look at the tradeoff between service level agreement (SLA) satisfaction and complex reward optimisation. We show how the fluid analysis naturally leads to a constrained global optimisation problem with embedded differential equations. We illustrate this problem on an abstract model of a virtualised execution environment that accurately captures resource allocations.
Hybrid performance modelling of opportunistic networks
 In: QAPL 2012, EPTCS 85
, 2012
"... We demonstrate the modelling of opportunistic networks using the process algebra stochastic HYPE. Network traffic is modelled as continuous flows, contact between nodes in the network is modelled stochastically, and instantaneous decisions are modelled as discrete events. Our model describes a netw ..."
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We demonstrate the modelling of opportunistic networks using the process algebra stochastic HYPE. Network traffic is modelled as continuous flows, contact between nodes in the network is modelled stochastically, and instantaneous decisions are modelled as discrete events. Our model describes a network of stationary video sensors with a mobile ferry which collects data from the sensors and delivers it to the base station. We consider different mobility models and different buffer sizes for the ferries. This case study illustrates the flexibility and expressive power of stochastic HYPE. We also discuss the software that enables us to describe stochastic HYPE models and simulate them.
Under consideration for publication in Formal Aspects of Computing HYPE: Hybrid modelling by composition of flows
"... Abstract. Hybrid systems are manifest in both the natural and the engineered world, and their complex nature, mixing discrete control and continuous evolution, make it difficult to predict their behaviour. In recent years several process algebras for modelling hybrid systems have appeared in the lit ..."
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Abstract. Hybrid systems are manifest in both the natural and the engineered world, and their complex nature, mixing discrete control and continuous evolution, make it difficult to predict their behaviour. In recent years several process algebras for modelling hybrid systems have appeared in the literature, aimed at addressing this problem. These all assume that continuous variables in the system are modelled monolithically, often with differential equations embedded explicitly in the syntax of the process algebra expression. In HYPE an alternative approach is taken which offers finergrained modelling with each flow or influence affecting a variable modelled separately. The overall behaviour then emerges as the composition of flows. In this paper we give a detailed account of the HYPE process algebra, its semantics, and its use for verification of systems. We establish both syntactic conditions (welldefinedness) and operational restrictions (wellbehavedness) to ensure reasonable behaviour in HYPE models. Furthermore we consider how the equivalence relation defined for HYPE relates to other relations previously proposed in the literature, demonstrating that our finegrained approach leads to a more discriminating notion of equivalence. We present the HYPE model of a standard hybrid system example, both establishing that our approach can reproduce the previously obtained results and demonstrating how our compositional approach supports variations of the problem in a straightforward and flexible way.
This work is licensed under the Creative Commons Attribution License. Patchbased Hybrid Modelling of Spatially Distributed Systems by Using Stochastic HYPE ZebraNet as an Example
"... Individualbased hybrid modelling of spatially distributed systems is usually expensive. Here, we consider a hybrid system in which mobile agents spread over the space and interact with each other when in close proximity. An individualbased model for this system needs to capture the spatial attribu ..."
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Individualbased hybrid modelling of spatially distributed systems is usually expensive. Here, we consider a hybrid system in which mobile agents spread over the space and interact with each other when in close proximity. An individualbased model for this system needs to capture the spatial attributes of every agent and monitor the interaction between each pair of them. As a result, the cost of simulating this model grows exponentially as the number of agents increases. For this reason, a patchbased model with more abstraction but better scalability is advantageous. In a patchbased model, instead of representing each agent separately, we model the agents in a patch as an aggregation. This property significantly enhances the scalability of the model. In this paper, we convert an individualbased model for a spatially distributed network system for wildlife monitoring, ZebraNet, to a patchbased stochastic HYPE model with accurate performance evaluation. We show the ease and expressiveness of stochastic HYPE for patchbased modelling of hybrid systems. Moreover, a meanfield analytical model is proposed as the fluid flow approximation of the stochastic HYPE model, which can be used to investigate the average behaviour of the modelled system over an infinite number of simulation runs of the stochastic HYPE model. 1
Noname manuscript No. (will be inserted by the editor) Meanfield Analysis of Hybrid Markov Population Models with Timeinhomogeneous Rates
"... the date of receipt and acceptance should be inserted later Abstract We consider a hybrid extension of Population Continuous Time Markov Chains (PCTMC) – a class of Markov processes capturing interactions between large groups of identically behaved agents. We augment the discrete state space of a P ..."
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the date of receipt and acceptance should be inserted later Abstract We consider a hybrid extension of Population Continuous Time Markov Chains (PCTMC) – a class of Markov processes capturing interactions between large groups of identically behaved agents. We augment the discrete state space of a PCTMC with continuous variables that evolve as integrals over the population vector and that can simultaneously provide feedback to the rates of transitions in the PCTMC. Additionally, we include timeinhomogeneous rate parameters, which can be used to incorporate real measurement data into the models. We extend meanfield techniques for PCTMCs and show how to derive a system of integral equations that approximate the evolution of means and higherorder moments of populations and continuous variables in a hybrid PCTMC. We prove first and secondorder convergence results that justify the approximations. We use a moment closure based on the normal distribution which improves the accuracy of the moment approximation in case of proportional control where transition rates depend on the amount a continuous variable is above or below a fixed threshold. We demonstrate how this framework is suitable for modelling feedback from globallyaccumulated quantities in a large scale system, such as energy consumption, total cost or temperature in a data centre. We present a model of a many server system with temperature management and external workload that varies with time. We show how to use real data to represent the workload within the framework. We use stochastic simulation to validate the example and an earlier example of a hypothetical heterogeneous computing cluster. 1